one-sided derivatives

•

If the real function f is defined in the point x0 and on some interval left from this and if the left-hand one-sided limitlimh→0-⁡f⁢(x0+h)-f⁢(x0)h exists, then this limit is defined to be the left-sided derivative of f in x0.

•

If the real function f is defined in the point x0 and on some interval right from this and if the right-hand one-sided limit
limh→0+⁡f⁢(x0+h)-f⁢(x0)h exists, then this limit is defined to be the right-sided derivative of f in x0.

It’s apparent that if f has both the left-sided and the right-sided derivative in the point x0 and these are equal, then f is differentiable in x0 and f′⁢(x0) equals to these one-sided derivatives. Also inversely.

Example. The real function x↦x⁢x is defined for
x≧0 and differentiable for x>0 with
f′⁢(x)≡32⁢x. The function also has the right derivative in 0:

limh→0+⁡h⁢h-0⁢0h=limh→0+⁡h=0

Remark. For a function f:[a,b]→ℝ,
to have a right-sided derivative at x=a with value d,
is equivalent to saying that there is an extensiong
of f to some open interval containing [a,b]
and satisfying g′⁢(a)=d. Similarly for left-sided derivatives.